Engineering Mathematics-I (GBTU/UPTU NAS-203)

✍ Scribed by Babu Ram

• Publisher: Pearson Education
• Year: 2013
• Tongue: English
• Leaves: 669
• Category: Library

✦ Table of Contents

Cover
Copyright
Dedication
Syllabus
Contents
Preface
1 Successive Differentiation and Leibnitz’s Theorem
1.1 Successive Differentiation
1.2 Leibnitz’s Theorem and Its Applications
2 Partial Differentiation and Expansion of Functions of Several Variables
2.1 Introduction
2.2 Continuity of a Function of Two Variables
2.3 Differentiability of a Function of Two Variables
2.4 The Differential Coefficients
2.5 Distinction Between Derivatives and Differential Coefficients
2.6 Higher-Order Partial Derivatives
2.7 Envelopes and Evolutes
2.8 Homogeneous Functions and Euler’s Theorem
2.9 Differentiation of Composite Functions
2.10 Transformation from Cartesian to Polar Coordinates and Vice Versa
2.11 Taylor’s Theorem for Functions of Several Variables
2.12 Miscellaneous Example
3 Asymptotes and Curve Tracing
3.1 Introduction
3.2 Determination of Asymptotes When the Equation of the Curve in Cartesian Form is Given
3.3 The Asymptotes of the General Rational Algebraic Curve
3.4 Asymptotes Parallel to Coordinate Axes
3.5 Working Rule for Finding Asymptotes of Rational Algebraic Curve
3.6 Intersection of a Curve and Its Asymptotes
3.7 Asymptotes by Expansion
3.8 Asymptotes of the Polar Curves
3.9 Circular Asymptotes
3.10 Concavity, Convexity and Singular Points
3.11 Curve Tracing (Cartesian Equations)
3.12 Curve Tracing (Polar Equations)
3.13 Curve Tracing (Parametric Equations)
4 Jacobian, Extreme Values and Applications of Partial Differentiation
4.1 Jacobians
4.2 Properties of Jacobian
4.3 Necessary and Sufficient Conditions for Jacobian to Vanish
4.4 Approximation of Errors
4.5 General Formula for Errors
4.6 Tangent Plane and Normal to a Surface
4.7 Differentiation Under the Integral Sign
4.8 Extreme Values
4.9 Lagrange’s Method of Undetermined Multipliers
5 Linear Algebra
5.1 Introduction
5.2 Concepts of Group, Ring, Field and Vector Space
5.3 Matrices
5.4 Algebra of Matrices
5.5 Associative Law for Matrix Multiplication
5.6 Distributive Law for Matrix Multiplication
5.7 Transpose of a Matrix
5.8 Transposed Conjugate of a Matrix
5.9 Symmetric, Skew Symmetric and Hermitian Matrices
5.10 Lower and Upper Triangular Matrices
5.11 Determinant of a Matrix
5.12 Adjoint of a Matrix
5.13 The Inverse of a Matrix
5.14 Methods of Computing Inverse of a Matrix
5.15 Rank of a Matrix
5.16 Elementary Matrices
5.17 Row/Column Reduced Echelon Form and Normal Form of Matrices
5.18 Rank of the Product of Matrices
5.19 Row and Column Equivalence of Matrices
5.20 Linear Dependence and Rank
5.21 System of Linear Equations
5.22 Solution of Non-Homogeneous Linear System of Equtions
5.23 Consistency Theorem
5.24 Homogeneous Linear Equations
5.25 Eigenvalue Problem, Eigenvalues, Eigenvectors and Characteristic Equation
5.26 Cayley–Hamilton Theorem
5.27 Algebraic and Geometric Multiplicity of an Eigenvalue
5.28 Orthogonal, Normal and Unitary Matrices
5.29 Similarity of Matrices
5.30 Diagonalization of a Matrix
5.31 Triangularization of an Arbitrary Matrix
5.32 Applications of Matrices to Engineering Problems
6 Beta and Gamma Functions
6.1 Introduction
6.2 Beta Function
6.3 Properties of Beta Function
6.4 Gamma Function
6.5 Properties of Gamma Function
6.6 Relation Between Beta and Gamma Functions
6.7 Dirichlet’s and Liouville’s Theorems
6.8 Miscellaneous Examples
7 Multiple Integrals
7.1 Introduction
7.2 Double Integrals
7.3 Properties of a Double Integral
7.4 Evaluation of Double Integrals (Cartesian Coordinates)
7.5 Evaluation of Double Integrals (Polar Coordinates)
7.6 Change of Variables in a Double Integral
7.7 Change of Order of Integration
7.8 Area Enclosed by Plane Curves (Cartesian and Polar Coordinates)
7.9 Volume and Surface Area as Double Integrals
7.10 Triple Integrals and Their Evaluation
7.11 Change to Spherical Polar Coordinates From Cartesian Coordinates in a Triple Integral
7.12 Volume as a Triple Integral
7.13 Miscellaneous Examples
8 Vector Calculus
8.1 Introduction
8.2 Differentiation of a Vector
8.3 Partial Derivatives of a Vector Function
8.4 Gradient of a Scalar Field
8.5 Geometrical Interpretation of a Gradient
8.6 Properties of a Gradient
8.7 Directional Derivatives
8.8 Divergence of a Vector-Point Function
8.9 Physical Interpretation of Divergence
8.10 Curl of a Vector-Point Function
8.11 Physical Interpretation of Curl
8.12 The Laplacian Operator ∇2
8.13 Properties of Divergence and Curl
8.14 Integration of Vector Functions
8.15 Line Integral
8.16 Work Done by a Force
8.17 Surface Integral
8.18 Volume Integral
8.19 Gauss’s Divergence Theorem
8.20 Green’s Theorem in a Plane
8.21 Stoke’s Theorem
8.22 Miscellaneous Examples
Examination Papers
Index